A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

learn more… | top users | synonyms

0
votes
0answers
9 views

Convolution to establish Gaussian process

A Gaussian process $z(s)$ can be established by convolving a gaussian white noise process $x(s)$ with a smoothing kernel $k(s)$ http://ftp.stat.duke.edu/WorkingPapers/01-03.pdf $$\\z(s)=\int_{S}^{} ...
0
votes
0answers
12 views

Problem with compound Poisson process

Let $X_k$ for $k=1,2,...$ be a sequence of i.i.d. random variables with $\mu_k=0$ and $\sigma_k^2=1$ for all $k$. Consider de random process $$S(t)=\sum\limits_{k=1}^{N(t)}X_k $$ where $N(t)$ is a ...
1
vote
2answers
17 views

Function evaluated in Brownian motion vanishes implies that the function itself vanishes?

tl;dr, here's my question: Question. Let $f(t,x)$ be a measurable function such that $f(t,B_t)=0$ almost everywhere on $[0,T]\times\Omega$ for a Brownian motion $B_t$. Does this imply that ...
0
votes
0answers
5 views

Identification of Infinite Dimensional State in Hidden Markov Model

Consider a hidden markov model (HMM) where the state, $X_t(\alpha)$, is a stochastic distribution over $\alpha \in \mathbb{R}_+$ and one observes a signal $Y_t$, which is simply a moment of this ...
0
votes
0answers
15 views

Can you identify this stochastic process?

So I run into this problem the other day and I cannot even think of the keywords I need to use to look it up. For the discrete random variable $X$ we have: $P_{\Delta X(t)} = F\big(X(t-1), ...
1
vote
0answers
7 views

Describe the law of a Bessel process conditioned on hitting $b$ before $0$

We are given the Bessel process SDE $$dX_t=\frac{\delta -1}{2X_t}dt+ dB_t, X_0>0.$$ Where $B_t$ is a standard Brownian motion, at least until $X_0=0$. We need to solve four problems: Show that ...
0
votes
0answers
27 views

Question regarding regular stochastic matrix

We say that a stochastic matrix is regular iff $\exists n\in \mathbb N$ such that $p_{ij}(n)>0$ for all states $i,j$ How many powers of a matrix do we need to compute at most in order to verify ...
1
vote
0answers
6 views

Distribution of an autoregressive process

Say that we are given a AR process. Also, lets assume that the residuals of the process come form a distribution $P_R$ which, while known to us, is not necessarily normal. Can I derive the ...
0
votes
1answer
23 views

What is the expectation of $\int_0^t \sqrt{s+B_s^2}dB_s$?

I am trying to find the expectation of $\int_0^t \sqrt{s+B_s^2}dB_s$, but am unable to use Ito's Formula because of the nasty integral. Is there another solution I am missing? Thanks!
-1
votes
0answers
10 views

How to find the derivative of $\int_0^t W_s^2 ds$, with respect to $W_s$, where $W_s$ is a Wiener process? [on hold]

I would like to find the derivative of $\int_0^t W_s^2 ds$, with respect to $W_s$, where $W_s$ is a Wiener process. Formally I want: $\frac{d}{dW_t}\int_0^t W_s^2 ds$. I understand that I can ...
4
votes
0answers
32 views

Stochastic domination

Suppose we have two probability measures on a space $X$, $\mu$ and $\nu$, such that $\nu$ stochastically dominates $\mu$, i.e.there exist a coupling of $\mu$ and $\nu$ on the product space $X \times ...
-3
votes
0answers
25 views

Proof for Yor's Formula [on hold]

Given that X and Y are two semimartingales, how can it be proved that the following statement is true? $\varepsilon(X)\varepsilon(Y)=\varepsilon(X+Y+[X,Y])$
0
votes
0answers
33 views

Why do we always consider real-valued $f$ in the Itō formula to find an expression for $f(t,X_t)$

The Itō formula (see Da Prato, Theorem 4.32) yields an expression for $f(t,X_t)$ where $${\rm d}X_t=\phi\;{\rm d}t+\Phi\;{\rm d}W_t\;,\;\;\;X_0=\xi\;.\tag 1$$ Even when $X$ takes values in a Hilbert ...
0
votes
0answers
4 views

Existence of first passage time density for time-inhomogeneous diffusion

Let $X$ be a time-inhomogeneous diffusion process in $\mathbb{R}^d$: $$dX_t=b(t,X_t)dt+\Sigma(t,X_t)dB_t,$$ where $\Sigma_{d\times d}$ is uniformly elliptic, and coefficients are such that the above ...
1
vote
0answers
14 views

How to find the mean of $\int_0^t W_s ds$, where $W_s$ is a Wiener process?

am trying to find the expectation of $\int_0^t W_s ds$, with $W_s$ being the Standard Wiener process. I am trying to use Ito's formula, by decomposing as: $$ \frac{W_t^3}{6} = \frac{1}{2}\int_0^t ...
1
vote
1answer
22 views

How to solve for the expectation of the Ito Integral: $\int_0^4 B_t^2 dB_t$?

I would like to find the expectation of the Ito Integral: $\int_0^4 B_t^2 dB_t$. My strategy is to use Ito's general formula with: $$ f(t, B_t) = f(0,0) + \int_0^t \frac{df}{dx}(s, B_s) dB_s + ...
1
vote
0answers
10 views

In stochastic calculus, what is the importance behind quadratic variation?

I am learning stochastic calculus right now and I came across several mentions of the computation of the quadratic variation of a Wiener process random variable. However, most of the resources I have ...
0
votes
0answers
6 views

For stochastic differential equations, why do we care if the process is $L^2$ bounded?

I have been studying Stochastic Differential Equations, and one theorem relates to the existence of a solution to the SDE: $$ dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dB_t $$ with $X_0 = x_0$ and $0 ...
2
votes
0answers
35 views
+50

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space ...
0
votes
0answers
7 views

Normal transitional pdf Wiener process continuity mistake and what other standard pdfs are used

I need you to tell me where I am making a mistake in the following: $$f_{1|1}(x_2,t+\Delta t|x_1,t) = \frac{1}{\sqrt{2\pi\Delta t}}e^-{\frac{(x_2-x_1)^2}{2\Delta t}}$$ If I let $\Delta x = x_2-x_1$, ...
2
votes
2answers
46 views

Showing that this is a martingale.(4.13 in Øksendals SDE)

This is an exercise from Øksendals stochastic differential equations, where I get stuck. It is exercise number 4.13.(I simplified the notation a bit.) I have that X is an Itô-process where: ...
0
votes
1answer
24 views

If voters arrive according to a Poisson process, how can we find the conditional number of votes of a candidate?

Suppose that voters arrive to a voting booth according to a Poisson process with rate $\lambda = 100$ voters per hour. The voters will vote for two candidates, candidate $1$ and candidate $2$ with ...
3
votes
0answers
34 views

Hitting Times for Brownian Motion - Levy Process?

Let $X$ be a Brownian motion and let $$H_a = \inf\{ s \ge 0 \mid X_s = a \} \;\ \text{and} \;\ S_a = \inf\{ s \ge 0 \mid X_s > a \}.$$ Now, I've shown that $H_a$ and $S_a$ are equal almost surely ...
0
votes
0answers
17 views

Working out payoff of a derivative with random interest rates

For this question, I've worked out the payoffs at N=3 but I'm not able to understand how to calculate the the expectation of the terms inside. If anyone could tell me how to find the expectation of ...
0
votes
0answers
13 views

Properties of Kernel Integral inner Product of Gaussian Process

Can anyone give any reference / suggest how to get the rigorous mathematical properties of the following : $$ Y=\int_{a}^{b} K_{X} (t) \ f(t) \ dt $$ where $$f \sim GP (\mu(\cdot), R ...
1
vote
0answers
27 views

Construction of a random variable

I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda. In Appendix A.2, where they discuss the construction of a random variable, there is the ...
0
votes
0answers
19 views

Estimate for average probability of Ito diffusion falls into an interval

Denote $E^x(X_t)$ be the solution to a Ito diffusion starting with $X_0=x$. Let $K\subset \mathbb{R}$ be a compact subset. I also assume $X^x_t$ has transition probability $p(t,y,x)$. Currently I am ...
-6
votes
0answers
77 views

Bernoulli Process [on hold]

Customers depart from a bookstore according to a Bernoulli process with parameter p = 0.15 (per minute). Each customer buys a book with probability 2/3, independent of everything else. Find the ...
1
vote
0answers
33 views

Some Kind of Generalized Brownian Motion

Let $\displaystyle X(t) = \int_0^t f(s)dB(s)$ where $B(t)$ is a Brownian motion and $f(t)\in L^2[0,1]$. What is a simple representation for $Y(t):=(X(t)|X(1))$ in terms of $B(t)$? Note, I am not ...
0
votes
1answer
21 views

brownian motion and process C1 (order 1 of continuity)

Here is my problem, With probability 1 (ie: a.s) the brownian motion $(B_t)_{t\in[0,T]}$ is continuous (which is define on a classic probability space $(\Omega, \mathcal{F}, ...
1
vote
2answers
36 views

Approximation of $\int_0^tF_x(s,X_s)Φ_0dW_s$ where $dX_s=φ_sds+Φ_sdW_s$ and $F_x$ is the Fréchet derivative of some $F:[0,t]×H→\mathbb R$

Let $U$ and $H$ be Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ be equipped with the usual inner product $(\Omega,\mathcal A,\operatorname P)$ ...
2
votes
0answers
15 views

Estimate for Expectation of Reciprocal Bessel Process

Let $W=(W_{t})_{t\geq 0}$ be a standard $3$-dimensional Brownian motion, and let $a\neq 0\in\mathbb{R}^{3}$. Consider the $3$-dimensional inverse Bessel process defined by ...
0
votes
1answer
10 views

Does an embedded discrete-time Markov chain preserve its properties in continuous time?

Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a continuous ...
1
vote
2answers
50 views

Itō formula as presented in “Stochastic Equations in Infinite Dimensions” by Giuseppe Da Prato

In Stochastic Equations in Infinite Dimensions, Theorem 4.32 (Google Books), the authors present the following version of an Itō formula: Given Hilbert spaces ...
4
votes
1answer
30 views

How to show that $(X(t)-\lambda t)^2 - \lambda t$ is a martingale, where $X(t)$ is a Poisson Process?

I am trying to show that $(X(t)-\lambda t)^2 - \lambda t$ is a martingale, where $X(t)$ is a Poisson process with rate $\lambda$. So far, what I have done is: \begin{align*} E\left((X(t)-\lambda t)^2 ...
2
votes
0answers
21 views

Expectation of an Exponentiated Integral of a Brownian Bridge

Given a Brownian bridge $X(t)$ where $X(0)=0$ and $X(1)$ equal to some given constant. What is $\displaystyle \mathbf E\Big[\exp\Big(\int_0^1X(t)dt\Big)\Big]$? I suppose I can always discretize the ...
1
vote
0answers
11 views

Gaussian filtering

I'm reading a paper and don't get how they tackle the drift of a gaussian process. We are in the setting of isonormal Gaussian processes. Let $Z$ be a Gaussian process with covariance operator ...
-1
votes
0answers
16 views

Symmetric Short Random Walk [on hold]

Let $\{S_n\}_{n\ge0}$ be a symmetric short random walk. Show that $$P_0 \left(\max_{0\le j\le n}|S_j | \ge b\right) \le 2 P_0\left(|S_n| \ge b\right)$$ *Sorry not sure how to type in Math symbols ...
0
votes
0answers
27 views

Integrability of a stochastic process

Let $x(t)$ be some random path $t\in[a,b]\subset\mathbb{R}$. I.e. $x:\Omega\rightarrow\mathbb{R}^{[a,b]}$ etc. When is $\int_a^b x(t)dt$ defined? If $x(t)$ is Brownian motion, I know it's ok. A ...
2
votes
0answers
21 views

I found $E(Σ_{j=0}^{k-1}η_j-Σ_{j=0}^{k-1}E(η_j|G_j))^2=Σ_{j=0}^{k-1}(E(η_j)^2-E(E(η_j|G_j)^2)$ in a book with faulty assumptions on the objects

In Stochastic Equations in Infinite Dimensions (Second Edition) on page 109, the authors state the following: If $\eta_0,\ldots,\eta_{k-1}$ are random variables with finite second moments and ...
2
votes
1answer
15 views

Bounding expectation of a supremum process

This is exercise 3.9(c) on page 15 of Karatzas and Shreve's Brownian Motion and Stochastic Calculus. Let $N_t$ be a Poisson process with intensity $\lambda$. In particular, if $t$ is fixed, $N_t$ is ...
1
vote
1answer
38 views

Spread of a rumor in a growing population

This is a variation on a classic problem. It occur's in several problems I am researching and I'd like to get some help from folks who may have dealt with this already or can offer insights. Let ...
1
vote
0answers
38 views

Diffusions: Conditional expectation of Stopping time

Out of Rogers - Williams: Diffusions, Markov Processes and Martingales: Page 279: Let $\{\mathbb{P}^x:x\in I\}$ be a regular canonical diffusion. For $y\in J:=[a,b]\subset I$ let $X_t$ denote the ...
1
vote
0answers
19 views

Intutitive understanding of the law of a stochastic process

I'm trying to get my head around the notion of "Law of a stochastic process" intuitively. This is what I got for a Brownian motion: Denoting the law of a Brownian motion ...
1
vote
0answers
25 views

Prove that a sum of random variables converges against an Itō integral

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace $f:[0,\infty)\times H\to\mathbb R$ be ...
2
votes
0answers
23 views

Derive an Itō formula for $f(t,X_t)$ where $X_t=X_0+tY+W_tZ$ and $f:[0,\infty)\times H\to\mathbb R$ and $H$ is a Hilbert space

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace $f:[0,\infty)\times H\to\mathbb R$ be Fréchet ...
2
votes
1answer
42 views

Is this an adapted process?(deterministic integrator in Itô-process)

Assume you have a probability space with a filtration, $(\Omega,\mathcal{F},P,\{\mathcal{F}_t\})$. Assume that the stochastic process $X_t$ is adapted to this filtration, and is jointly measurable ...
3
votes
1answer
39 views

Example of a Continuous-Time Markov Process which does NOT have Independent Increments

1. Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a ...
0
votes
0answers
7 views

How to normalize sets of scores to have very similar histogram?

I have the output of several stochastic processes I need to combine into a single value. They have similar histogram curves, but not exactly the same. These curves are not perfectly Gaussian (see ...
1
vote
0answers
16 views

“Local” functional central limit theorem for the empirical distribution function

Assume $(X_i)_{i=1}^{\infty}$ is a sequence of i.i.d. real-valued random variables such that $\mathbb E[X^2]<\infty$. Denote by $F_X(t) := \mathbb P(X\leq t)$ their common distribution function. ...